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added rough description and original citation to Adams e-invariant
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added the original reference for the interpretation in bordism theory:
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and this one:
I have added statement and proof (here) of the e-invariant as the top degree component of the Chern character on the cofiber space:
$\exp \left( 2 \pi \mathrm{i} \int_{C_f} ch\big( V_{2n} \big) \right) \;=\; \exp \left( 2 \pi \mathrm{i} \, { \color{blue} e(f) } \right) \;\;\; \in \mathrm{U}(1)$added the statement (here) that the e-invariant is Todd class of cobounding (U,fr)-manifolds
Am starting a new experimental section “Construction via unit cofiber cohomology theories” (here) meant to lay out another approach to constructing the Adams e-invariant, more abstractly homotopy theoretic and maybe not considered in the literature (?).
If it works out, this is such that it makes various facts immediately manifest, notably the equality between Adams’ construction via the Chern character on KU with Conner-Floyd’s construction via the Todd character on MUFr.
So far the section contains one Lemma, identifying the “unit cofiber cohomology” of the cofiber space under even-periodic ordinary cohomology.
From this, the whole story should follow from looking at a single homotopy pasting diagram, to be included in a moment.
What’s that link in #16? I can’t see it even from the source code of the page.
The link in #16 is meant to go to theorem 5.11 in Adams e-invariant. It points to the anchor named DiagrammaticeCInavriantReproducedClassicaleCInvariant
, which is the label of that theorem.
I just checked again, clicking on it, and it works for me. What happens on your side when you click in #16?
Maybe this is a caching issue: If you had the page already opened, but with an earlier version loaded, then maybe your browser looks for the anchor in the old page without reloading, and then doesn’t find it.
Ah yes, it works now.
I have fixed the statement in this Prop. of the entry (currently Prop. 5.6)
(i.e. the generalization to any multiplicative cohomology theory of the Conner-Floyd construction of a $U$-cobordism class with framed boundary from the trivialization of a d-invariant in $M U$-theory).
Namely, it used to say that the construction lifts through the boundary map as a bijection. But that’s evident nonsense:
It is a bijection if we retain the information of the 2-homotopy class of the homotopy involved in the cone shown in the first version of the proof. But as we don’t retain that information (one could, but it’s besides the point here) it’s just a map that lifts through $\partial$.
I have made a brief fix. But I realize there is room to beautify the statement of the proposition. Maybe later.
An analogous comment and fix applies to the corresponding proposition at d-invariant
After the statement/proof that the $e_{\mathbb{C}}$-invariant defined in terms of Adams operations equals the top degree coefficient of the Chern character
I have added a warning (now this Remark) that the analogous statement for $e_{\mathbb{R}}$ in general fails.
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